If the conics whose equations are `S-=sin^2thetax^2+2h x y+cos^2thetay^2+32 x+16 y+19=0,S^(prime)-=cos^2thetax^2+ ` `2h^(prime)x y+sin^2thetay^2+16 x+32y+19=0` intersect at four concyclic points, then, (where `theta in R)` (a) `h+h^(prime)=0` (b) `h=h '` (c)`h+h^(prime)=1` (d) none of these

If the conics whose equations are S_1:(sin^2theta)x^2+(2htantheta)x y+(cos^2theta)y^2+32 x+16 y+19=0 S_2:(cos^2theta)x^2-(2h^(prime)cottheta)x y+(sin^2theta)y^2+16 x+32 y+19=0 intersect at four concyclic points, where theta[0,pi/2], then the correct statement(s) can be (a) h+h^(prime)=0 (b) h-h^(prime)=0 (c) theta=pi/4 (d) none of these

The equation a^2x^2+2h(a+b)x y+b^2y^2=0 and a x^2+2h x y+b y^2=0 represent

If two curves, whose equations are ax^2+2hxy+by^2+2gx+2fy+c=0 and a'x^2+2h'xy+b'y^2+2g'x+2f'y+c'=0 intersect in four concyclic points, prove that (a-b)/h=(a'-b')/(h')

If the pair of lines a x^2+2h x y+b y^2+2gx+2fy+c=0 intersect on the y-axis, then prove that 2fgh=bg^2+c h^2

The condition that one of the straight lines given by the equation a x^2+2h x y+b y^2=0 may coincide with one of those given by the equation a^(prime)x^2+2h^(prime)x y+b^(prime)y^2=0 is (a b^(prime)-a^(prime)b)^2=4(h a^(prime)-h^(prime)a)(b h^(prime)-b^(prime)h) (a b^(prime)-a^(prime)b)^2=(h a^(prime)-h^(prime)a)(b h^(prime)-b^(prime)h) (h a^(prime)-h^(prime)a)^2=4(a b^(prime)-a^(prime)b)(b h^(prime)-b^(prime)h) (b h^(prime)-b^(prime)h)^2=4(a b^(prime)-a^(prime)b)(h a^(prime)-h^(prime)a)

If f^(prime)(x)=sqrt(2x^2-1) and y=f(x^2),t h e n(dy)/(dx) at x=1 is (a)2 (b) 1 (c) -2 (d) none of these

If (a x^2+b x+c)y+(a^(prime)x^2+b^(prime)x+c^(prime))=0 and x is a rational function of y , then prove that (a c^(prime)-a^(prime)c)^2=(a b^(prime)-a^(prime)b)xx(b c^(prime)-b^(prime)c)dot

If one of the lines denoted by the line pair a x^2+2h x y+b y^2=0 bisects the angle between the coordinate axes, then prove that (a+b)^2=4h^2

If sin^(-1)x+sin^(-1)y=pi/2,t h e n(1+x^4+y^4)/(x^2-x^2y^2+y^2) is equal to 1 (b) 2 (c) 1/2 (d) none of these

Given the function f(x)=(a^x+a^(-x))/2(w h e r ea >2)dotT h e nf(x+y)+f(x-y)= (A) 2f(x).f(y) (B) f(x).f(y) (C) f(x)/f(y) (D) none of these